Question

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function \(f\) on \([a, b] .\) Identify \(f\) and express the limit as a definite integral. $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(4-x_{k}^{* 2}\right) \Delta x_{k} \text { on }[-2,2]$$

Short answer

Question: Express the limit of the given Riemann sum as a definite integral: $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(4-x_{k}^{* 2}\right) \Delta x_{k} \text { on }[-2,2].$$ Answer: The definite integral equivalent to the given Riemann sum is $$\int_{-2}^{2}(4-x^2)dx.$$

Step-by-step solution

Identify the function \(f\) and interval of integration

The function \(f\) is given by the expression inside the sum, which is \(4-x_{k}^{* 2}\). The interval of integration is provided as \([-2, 2]\).

Write the limit of the Riemann sum as a definite integral

The limit of the Riemann sum as \(\Delta \rightarrow 0\) can be written as a definite integral. For our given limit, $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(4-x_{k}^{* 2}\right) \Delta x_{k} \text { on }[-2,2],$$ we can express it as a definite integral on the interval \([-2, 2]\) as follows: $$\int_{-2}^{2}(4-x^2)dx.$$ So, the required definite integral is: $$\int_{-2}^{2}(4-x^2)dx.$$